Trouble with understanding section of FOL completeness proof

[pooj4]

The Completeness Proof for First-Order Predicate Logic depends on if $\Phi$ is a
set of consistent $\mathcal L$-formulas, then $\Phi$ is satisfiable.

How is that constructed? There are a large number of Lemmas working from Machover's text Set theory, Logic and Their Limitations but I'm having trouble with which are most relevant and how it comes together.

 

[Evgeny.Makarov]

The first step is to turn a consistent set $\Gamma$ into maximally consistent set $\Gamma^*$, i.e.., for each formula $A$ add either $A$ or $\neg A$ to $\Gamma$ in a way that preserves consistency. Then one defines an interpretation $I$ where the domain is the set of all closed terms and $I\models P(t_1,\ldots,t_n)$ if $P(t_1,\ldots,t_n)\in\Gamma^*$. Finally one proves that $I\models A\iff A\in\Gamma^*$ for all formulas $A$, not just atomic. Therefore $I\models\Gamma^*$ and in particular $I\models\Gamma$ since $\Gamma\subseteq\Gamma^*$.

I recommend starting with the last step. Try proving $I\models A\iff A\in\Gamma^*$ by induction on $A$ and see what this requires of $\Gamma^*$. Some steps go through for an arbitrary $\Gamma^*$; others require properties like completeness or existential completeness, which are ensures by the lemmas you mentioned. Also the book "The Open Logic Text" (Complete Build) has an outline of the proof in section 19.2.